Gate-tunable anomalous Hall effect in Bernal tetralayer graphene

Large spin-orbit coupling is often thought to be critical in realizing magnetic order-locked charge transport such as the anomalous Hall effect (AHE). Recently, artificial stacks of two-dimensional materials, e.g., magic-angle twisted bilayer graphene on hexagonal boron-nitride heterostructures and dual-gated rhombohedral trilayer graphene, have become platforms for realizing AHE without spin-orbit coupling. However, these stacking arrangements are not energetically favorable, impeding experiments and further device engineering. Here we report an anomalous Hall effect in Bernal-stacked tetralayer graphene devices (BTG), the most stable configuration of four-layer graphene. BTG AHE is switched on by a displacement field and is most pronounced at low carrier densities. The onset of AHE occurs in tandem with a full metal to a broken isospin transition indicating an orbital origin of the itinerant ferromagnetism. At lowest densities, BTG exhibits an unconventional hysteresis with step-like anomalous Hall plateaus. Persisting to several tens of kelvin, AHE in BTG demonstrates the ubiquity and robustness of magnetic order in readily available and stable multilayer Bernal graphene stacks—a new venue for intrinsic non-reciprocal responses.


5)
In the band structure calculation, the authors assume that the two graphene layers in the middle (Supplementary equation 4) remain neutral and use Ref. 5 to support this assumption.However, Ref. 5 is for a trilayer graphene such that it is general to set (V/2, 0, -V/2) from top to the bottom layers by assuming a homogeneous electric field distribution across the layers.In fact, for four-layers, generally, (3V/2,V/2,-V/2,-3V/2) or (V/2,V/6,-V/6,-V/2) is used.Especially near zero density, electric field cannot be screened so perfectly that only the top-most and bottom-most layers feel the potential V from the gate (you can check few literatures like PRB 102 035421).Authors either need to calculate the band structure and possibly also the Berry curvature again or provide stronger arguments why the Supplementary equation ( 4) can be used.6) To help readers quickly find the relevant figures in the SI, please specify the figure number when referring to them in the main text.7) There are few typos and mistakes in the paper.Please check the manuscript and supplementary carefully again.Here are few that I found: i) At page 2: there is an extra minus sign in 'The carrier density n = [Ct(Vt-Vt0-) + Cb(Vb-Vb0)]/e', ii) At page 4: authors wrote 'Total Hall resistance (without subtraction of linear background) at D=0.4 V/nm …' when they refer to Fig. 4a but the figure shows RAH, iii) the information about the magnetic fields seem missing in Figs.4f-h, iv) The reference 19 has two papers.
Reviewer #2 (Remarks to the Author): I co-reviewed this manuscript with one of the reviewers who provided the listed reports.This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.
Reviewer #3 (Remarks to the Author): The authors report an anomalous Hall effect (AHE) and magnetic hysteresis in Bernal tetralayer graphene.The data are high quality and the flow of the text and claims are clear.While high-quality samples of Bernal tetralayer graphene have been explored with low-temperature transport before (e.g.Refs. 15,16), the authors here focus on the behavior of their samples at low carrier densities, with finite electric displacement field, and low/zero magnetic field.In that regime, they find evidence of spin/valley flavor-symmetry breaking and the primary result, anomalous Hall resistance and hysteresis.
They ascribe the emergence of anomalous Hall resistance to net orbital magnetism developing within a "valley metal" region of the low-temperature phase diagram.Interestingly, they observe that the AHE persists to several tens of kelvin, a surprisingly high temperature given the weak magnetic hysteresis observed at the lowest temperatures, and given similar anomalous behavior observed in rhombohedral trilayer graphene only at lower temperatures.Another intriguing result is the development of a series of plateaux in the Hall resistance as a function of magnetic field, though the authors do not comment extensively on the latter.
Overall, their argument for a net valley imbalance and orbital magnetism is convincing, aside from some details that I will outline below.The results are timely and relevant to the discussion of related effects in moire and non-moire graphene systems such as Bernal and rhombohedrally stacked multilayer graphene.After addressing the following points, I believe the manuscript will be suitable for publication in Nature Communications.
1.The authors focus on the behavior of electrons in the manuscript, but they do not address whether they observed any anomalous Hall signatures for holes.Do they only see a linear Hall component at low densities and large displacement fields for holes?According to their DOS calculation (Fig. 1b), the Van Hove singularities in the valence band appear at least as large if not larger than in the conduction band.Naively, one might therefore expect symmetry breaking for holes as well as electrons (and possibly an AHE depending on the flavor polarization).Do they see evidence of symmetry breaking in the quantum oscillations for holes? 2. The boundary between the full and valley metals shown in Fig. 2a (gray line) appears quite sharp, but the onset of anomalous Hall resistance and hysteresis is gradual as a function of density near that boundary (e.g.Fig. 3e).Do they authors have a simple explanation for that difference?Does the density-or energy-dependence of the calculated Berry curvature account for the smooth onset of anomalous Hall? 3. Along those lines, what is the lower density limit for the observed AHE?Does the hysteresis loop close suddenly as the chemical potential approaches the gap or is there a gradual decrease in the AHE at densities below n=0.1?4. The step-like features observed in Rxy at low temperatures are intriguing, but to rule out possible domain physics, did the authors measure similar steps in additional devices?Supplementary Fig. 9 does not seem to show such steps.In the primary device shown in the main text, did the authors check if the steps are consistent between multiple contact pairs?If they result from spatial inhomogeneity rather than something more exotic, like momentum polarization, different contact pairs may show steps occurring at different coercive fields.5. Do the authors have a map of Rxy measured at B=0?This may help identify at a glance where the AHE is most pronounced.
Minor corrections: 6.In Fig. 1b, it would be informative to include real DOS units rather than arbitrary units.This will aid in comparing the magnitude of the Van Hove singularities to other related systems (Bernal, rhombohedral, moire, etc).
7. The color scale and/or color map used in Fig. 1a should be selected to show the quantum oscillations in the low-density ("VM") regime more clearly since this region is the focus of the manuscript.
8. The curves in Fig. 2b are not labeled according to where they were extracted from Fig. 2a (the caption mentions "corresponding symbols" but does not show them).9.It would be much easier to understand the AHE behavior shown in Fig. 3 if there were an inset of the map shown in Fig. 2a with corresponding points or lines superimposed to show the location of each hysteresis loop.An alternative would be to add matching labels in Fig. 2a, but this may appear quite busy since there are already labeled points in that map for the FFT analysis.10.I suggest doing something similar in Fig. 4, adding labels for the curves shown in Fig. 4a-e in the map shown in Fig. 4f.
11.In the final paragraph, the authors mention "magnetic field-free addressability of the valleys" in in BTG.In what sense are the valleys addressable at zero magnetic field?Unless the authors have evidence of magnetic switching with electric control (e.g.current, density, or displacement field switching), it seems that a magnetic field is required to switch and address different valley-polarized states.
12. Ref. 9 seems out of place in the abstract sentence, "Here we report an anomalous Hall effect in Bernal-stacked tetralayer graphene devices (BTG), the most stable configuration of four layer graphene [7][8][9]."In addition, it would make sense to cite Refs.15,16 in that sentence.13.Refs.19,20 should be added to the sentences, "...these enhance interactions to facilitate ferromagnetic ordering and a Stoner-type lowering of the four-fold isospin degeneracy [6,[22][23][24][25].Van Hove driven symmetry breaking in two and three layer graphene have recently been found to yield a myriad of spin, valley, and other flavor polarized states [6,[22][23][24][25][26]." Reviewer #4 (Remarks to the Author): I co-reviewed this manuscript with one of the reviewers who provided the listed reports.This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.
Reviewer #5 (Remarks to the Author): The authors of this work demonstrate the existence of an anomalous hall effect in tetralayer graphene in the presence of a finite magnetic field and finite displacement field.The result is important and the measurements seem reliable.I would like the authors to reply to the following questions before allowing for publication of the paper.
1) The authors should rationalize better the reason why the AHE occurs only at low carriers concentration and high-displacement field.In particular, I am not so convinced by the need of an high displacement field.Is it due to the experimental difficulties in detecting the effect or is there any conceptual reason for which we should have a large enough displacement field ?
Can the authors explain this better by looking at previous theoretical works on the subject or by using their 8 bands TB model ?
2) An half metallic state associated with a broken symmetry state breaking time reversal was also detected recently in few layers rhombohedral graphene ( see for example https://doi.org/10.1021/acs.nanolett.2c00466).In that case it was attributed to a layer antiferromagnetic state.Could it be that some kind of magnetic state is present also here (with a weak magnetization) ?Can the author exclude in some way that the broken symmetry state is magnetic ?

Point-by-Point Responses
Reviewer #1 (Remarks to the Author): In this work, the authors carried out magnetoresistance measurements in dual-gated Bernal-stacked tetralayer graphene (BTG).Using the dual-gated structure, authors tuned displacement field (D) and charge density (n) independently and observed anomalous Hall effect (AHE) at large displacement field (D) and low carrier density (n) in the region labelled as a valley metal ('VM').This is striking because it suggests that the time-reversal symmetry is spontaneously broken at zero magnetic fields.Moreover, quantum oscillation analysis shows that the total degeneracy is two in the VM region (a half metal) instead of one (so-called a quarter metal), indicating that the observed AHE is different from those found in rhombohedral trilayer and twisted bilayer graphene aligned with hBN.To find out the origin and support their claim, authors calculated Berry curvature of the BTG band at finite D near zero density and showed that non-zero Berry flux exists near zero energy.However, since the Berry flux is opposite in different valleys, to explain their observation, the valley degeneracy has to be broken.The authors claim that this is from the strong interactions at the van Hove singularity point that presents near zero energy coexisting with a finite Berry flux.
The work is interesting and shows that Bernal-stacked graphene multilayers can exhibit many-body physics even without further engineering like twisting or changing stacking order.However, there seem some weak points that need to be addressed before being accepted for publication in Nature Communications.Here are my comments and questions:

Response:
We thank the reviewer for recognizing the novelty and value of our work.We also thank the reviewer for providing constructive comments and suggestions which we have fully adopted and addressed.We have prepared a point-by-point response to these comments below including appropriate changes in the manuscript and supplementary information.All changes have been highlighted in yellow.We believe these changes significantly improve the quality of our work and address the questions of the reviewer.
1) In previous study on BTG (Ref. 15), Landau level sequence (or the lowest Landau level observed at the lowest magnetic fields) was also checked to confirm the stacking order and number of layers.Although Figs. 1c and 2a look quite close to those shown in Ref. 15, it would be good to show the quantum-Hall effect data in Supplementary Information or mark the corresponding Landau level filling factors for the vertical stripes in Fig. 2a such that the readers can check the Landau level sequence.In the current form, there seems no clear way for the readers to check the Landau level filling factors.

Response
We thank reviewer for this constructive suggestion.As suggested by reviewer and to make reading the Landau levels clearer, we have re-plotted the Fig. 2a with filling factors  labelled in the up x-axis (shown below as Figure R1 for the reviewer's convenience).Here =nh/eB, where n, h, e and B denotes carrier density, Planck constant, elementary charge and magnetic field, respectively.This replotted Fig. R1 demonstrates the Landau level sequences clearly.Additionally, to avoid an overcrowded Fig. 2a, we have directly placed this replotted figure in the Supplementary Information as suggested by the reviewer as Supplementary Fig. 2; please see also updated SI.We have also added a sentence in the main text that points the reader to Supplementary Fig. 2 to see the Landau level sequence.2) According to the calculation shown in Fig. 1b and Supplementary Fig. 6b, the hole band seems to be flatter with a larger density of states.However, in the analysis, authors didn't present the corresponding data nor discuss it.In fact, in Fig. 2a, the hole side does not exhibit a clear stich-like ripple pattern (i.e., the signature of the VM region).It would be interesting to see if this contrast is intrinsic or comes from the sample quality.For instance, since the band dispersion is different on the hole side, Berry flux may remain zero nor valley degeneracy is not broken.

Response:
We thank the reviewer for pointing out the faint stitch-like ripple pattern on hole side.
We agree that the resistance plot is not very well resolved on the hole side.This lower resolution on the hole side was also observed in previous experiments on Bernal tetralayer graphene, e.g., figure 4a  Of course, we agree with the reviewer that the hole side can also manifest regions of large density of states and energy windows with flat-like bandstructure.This clearly motivates the reviewer's interesting question about whether spontaneous Valley Symmetry breaking also manifests in the hole side.Indeed (as anticipated by the reviewer), we find that a hysteretic AHE emerges on the hole side, see Figure R2 plotted below for the convenience of the reviewer.A hysteretic AHE is a clear tell-tale signature of the broken valley symmetry in BTG which, as we now see, also appears in both electron and hole sides.Figure R2 and a short description can be now found in Supplementary Fig. 6 and Supplementary Section 5, respectively.We have also modified the main text to describe that hysteretic AHE signatures persist in the hole side as well, and have pointed to Supplementary Section 5 where the AHE data for the hole side is presented.
Similar to electron side, the hysteretic AHE curve in the hole side becomes more pronounced at lower hole density (see Fig. R2a-d); AHE seems to vanish for hole densities beyond 10 12 cm -2 hole.While the hole side does exhibit some knee-like step features (Figure R2a) they are less pronounced than what we observed for the electron side (Figure 3). 3) Authors claim that in FM region, AHE disappears and shows some AHE data measured at different D and n.However, it is not straightforward to check the crossing points, for instance, in Fig. 2a.It would be good to mark the positions in (n,D) at which the AHE disappears in Fig. 2a or somewhere the authors prefer.Alternatively, authors can add the points in Fig. 1c as it is measured at zero magnetic fields where AHE appears.

Response:
We thank the reviewer for pointing this out and for the excellent suggestion to improve the readability of the manuscript.As suggested by the reviewer, we have now added add black cross symbols ('X') directly in Fig. 1c to mark the (n, D) conditions corresponding to those in Fig. 3e.As can be seen, the AHE (almost) disappears in the top trace of Fig. 3e (corresponding to n=1.0 and D =0.3).This (n=1.0,D=0.3)condition corresponds to the right most cross symbol (X) in Fig. 1c.All labels have been described in both the captions of Figure 1 as well as Fig. 3. 4) Related to the question #3, it seems that at n=0.1, where the authors present most of the AHE data, the longitudinal resistance measured at finite magnetic fields (see, e.g., Figs.2a and 4f) shows a clear dip accompanied by more than two orders of magnitude larger resistance nearby.What is the origin of this state?Is it confirmed that this state is 'VM'?

Response:
We thank the reviewer for pointing out this issue.To clarify, the color bar in Fig. 2a and Importantly, in our work we have concentrated on valley symmetry breaking and the reporting the observation of anomalous Hall effect in BTG (i.e. at B=0).In inversion broken BTG, where the valleys have opposite signs of Berry curvature, a hysteretic anomalous Hall effect is a tell-tale signature of valley symmetry breaking.Indeed, in the n=0.1 region, we find that the AHE in BTG becomes more pronounced as compared with higher densities, see e.g., Fig. 3e of the main text.This is a clear signature of spontaneous valley symmetry breaking.Further, the relatively low resistance of this state ~ 200-300 ohms at zero magnetic field indicates a metallic character of the transport in BTG in this region.As such, we can confirm that the state around the n=0.1 region is a spontaneous valley symmetry broken state -i.e., VM state.5) In the band structure calculation, the authors assume that the two graphene layers in the middle (Supplementary equation 4) remain neutral and use Ref. 5 to support this assumption.However, Ref. 5 is for a trilayer graphene such that it is general to set (V/2, 0, -V/2) from top to the bottom layers by assuming a homogeneous electric field distribution across the layers.In fact, for four-layers, generally, (3V/2,V/2,-V/2,-3V/2) or (V/2,V/6,-V/6,-V/2) is used.Especially near zero density, electric field cannot be screened so perfectly that only the top-most and bottom-most layers feel the potential V from the gate (you can check few literatures like PRB 102 035421).Authors either need to calculate the band structure and possibly also the Berry curvature again or provide stronger arguments why the Supplementary equation ( 4) can be used.Part 2: Importantly, the distribution of the Berry curvature plays a significant role in the phenomenology of the AHE we observe.To see this, we first note that Berry curvature distribution in each valley is concentrated at the band edge (see Fig. 3c of main text).
As a result, the sharpest change in Berry flux (as a function of chemical potential) occurs close to the band bottom too (see Fig. 3d of main text).As a result, even small valley polarizations ( f K (p) ¹ f K' (p) ) when chemical potential is fixed close to the band bottom (i.e.low density), can result in pronounced AHE.In contrast, when chemical potential is large (i.e. at high density) the change of Berry flux with chemical potential is small, making it harder to detect an AHE for modest valley polarizations.Second, we note that Berry curvature in each of the valleys depends on the application of a finite displacement field; Berry curvature vanishes when there is no displacement field.For very small displacement fields, the Berry curvature is highly concentrated at the band bottoms making access to the region with sharp change in Berry flux challenging.As a result, having a modest strength of displacement field is needed.This is consistent with our observation that AHE is turned on by the displacement field (see e.g., Supplementary Fig. 4 as well Supplementary Figure 8 of our revised manuscript) and is most pronounced for low densities.
To make the connection between the valley polarization and the observed AHE phenomenology in our devices stronger, we have included a discussion of how the observations of AHE being enhanced at low density and being turned on by D field is directly connected and is consistent with the valley polarization and D-field induced Berry curvature distribution in BTG.This is now shown in the main text.
2) An half metallic state associated with a broken symmetry state breaking time reversal was also detected recently in few layers rhombohedral graphene ( see for example https://doi.org/10.1021/acs.nanolett.2c00466).In that case it was attributed to a layer antiferromagnetic state.Could it be that some kind of magnetic state is present also here (with a weak magnetization) ?Can the author exclude in some way that the broken symmetry state is magnetic ?
Response: We thank the reviewer for the insightful question and the useful reference.
We would like to answer the answer in two parts in which we will compare the main features of the magnetotransport data and magnetic state in Y.

Fig. R1 .
Fig. R1.Carrier density and displacement field dependence of four probe resistance under B=1T.The lower x-axis labels carrier density and the upper x-axis label display the filling factors =nh/eB, respectively.Here we have concentrated on the electron side where the oscillations are most pronounced.

Fig. R2 .
Fig. R2.Hysteresis and anomalous Hall curves of hole side, with different carrier density conditions at fixed D=0.3 V nm -1 .n and D are in units of 10 12 cm -2 and V nm -1 , respectively.Here the negative n values indicate hole density.

Fig. 4f are
Fig. 4f are in ln-scale (natural logarithm, with base e~2.718)instead of log-scale (with base 10), and therefore the BTG does not have a drastic resistance change near the n=0.1 region.Instead, the change in resistance is e^2 or by a factor of about 7.This convention is indicated by the label "ln R xx " in Figure 1 and 2 of the main text.Importantly, this resistance change does not occur at B=0 (see Fig. 1c of main text); it only occurs at finite B. To draw conclusions about the VM state, it is important to study the B=0 behavior.

Figure
Figure3cin main text simulated from our 8 band model.The Berry curvature sign is opposite for opposite valleys (K vs K').As a result, when time-reversal symmetry is preserved, the total net Berry flux (i.e.Berry flux in valley K + Berry flux in valley K') of the material vanishes and no anomalous Hall effect is detected.
of high density of states, valley symmetry can be spontaneously broken (e.g., through a Stoner-like ferromagnetism) where the distribution function in valley K and K' now become different: the state is valley polarized f K (p) ¹ f K' (p).As a result, that Berry flux in valley K and valley K' are now imbalanced leading to a finite total net Berry flux and an anomalous Hall effect.
In our work, we study Bernal stacked tetralayer graphene (BTG) using a Hall-bar set-up.Spontaneously broken time-reversal symmetry is confirmed by a hysteretic AHE (with out-of-plane magnetic field) and is supported by quantum oscillations measurements.The main features of the state we find in our BTG devices are If the graphene multilayer structure is only spin polarized, an AHE is not expected.Indeed, a four-terminal measurement of a spin polarized half-metallic state in rhombohedral trilayer graphene was performed in Nature 598, 429-433 (2021) revealing a vanishing anomalous Hall effect.In summary, a finite AHE is enabled only if valleys are distinctly populated (valley polarization).C3.If in addition to broken valley symmetry, spin is also polarized, quantum oscillations would show a quarter metal state.However, in the VM region, our quantum oscillations do not show evidence of a quarter metal state.Instead our quantum oscillations are consistent with a half metallic state (as discussed in the main text and shown in Figure2).C4.Additionally, we find that the AHE hysteresis in BTG persists to several tens of kelvin (see Figure4in main text).Whereas the hysteresis in two-terminal magnetoconductance disappear at a few Kelvin scale see (Y Lee et al., page 3 left column as well as abstract).This provides some indication that the energy scales and physics involved in the two devices are distinct.C5:It is pointed out by Y. Lee et al. (see page 2, left column) that the magnetoconductance features they find are exclusive to rhombohedral stacks being minimal in Bernal stacks that they made.On the basis of these comparisons (C1-C5) and on our experimental observations (e.g., BTG 1,2,3) we conclude that we have no evidence for spin polarization in our BTG devices.Further, the experimental phenomenology seen by Y. Lee et al. is quite different from what we see in BTG (we see no correlated gap, AHE is turned on at finite displacement field, hysteresis persists to several tens of kelvin).As a result, we also have no experimental evidence for an LAF state in our devices.Instead, our measurements are most consistent with that of a spontaneously broken valley symmetry state and itinerant ferromagnetism.It can be understood (as has been the case with a number graphene multilayers and moiré systems, e.g.,Ref.6,19,[22][23][24]ofourmain text) via a phenomenological Stoner ferromagnetism framework.To make the distinction between RFG and BTG clear, we have added a short description of the phenomenology of RFG spin polarized half metals in both Y.Lee et al.Nano Lett.22,5094-5099 (2022) [our Ref. 28] and Nature 598, 429-433 (2021) [our Ref. 6].
state is moderately doped (see figure 4b in Y. Lee et al).Note that spin-polarized half metals in rhombohedral trilayer graphene do not possess an AHE [as explicitly measured in Nature 598, 429-433 (2021)] BTG: